A. Anomalous Critical Behavior
The "critical" behavior of the viscosity can be inferred from Figures 28 and 29 for xenon and Figures 32 and 33 for ethane. Away from the critical density, the viscosity isochores are relatively flat until the coexistence curve is reached. Close to critical density, however, the viscosity isochores are flat only until the critical tern- perature is approached. At this point, the viscosity increases as the temperature is lowered. This increase can be considered to be
"anomalous" critical behavior. The range of the data may be too nar- row for the isothermal representation to show the character of the anomaly, but the critical increase is quite apparent. The behavior of the viscosity seems to be similar to that found by Naldrett and
[12] . [13]
Maass and Kestin, et al. .
It is of interest to consider the behavior of the "anomalous"
viscosity as a function of reduced temperature difference £ along the critical isochore. As previously discussed, the relationship of properties with reduced temperature difference is a convenient way to characterize the critical behavior. In particular, Eqs. (3) and (4) which have been used to characterize the viscosity of the binary
liquid mixtures can be similarly used for single component viscosities.
One of the difficulties in the use of these relationships is that the characteristic behavior is of ten strongly affected by the nature of the anomalous viscosity term.
J. V. Sengers[Sl] suggests the following form for the anomalous viscosity along the critical isochore 6n :
where
6n = n(p c ,T) - n(P c ,T f) re + n(O,T f) -re n(O,T) is the measured viscosity
n(p ,T f) is the viscosity along the critical isochore c re
at a super-critical reference temperature
n(O,T f) is the dilute gas viscosity re at zero density and the reference temperature
(75)
n(O,T) is the dilute gas viscosity at the temperature of measurement.
Equation (75) is derived as follows. The anomalous viscosity can be represented as:
6n
=
n(P c ,T) - n.d{p ,T)l. c (76)
where is the "ideal" viscosity which would be the value of the viscosity in the absence of any critical behavior. Consider the non- critical excess viscosity n* , which is the difference between the actual viscosity and the dilute gas viscosity at the same temperature:
(77)
It is often assumed that the excess viscosity is a function of density alone and independent of the temperature[SOJ. In the absence of critical effects, the viscosity in the critical region would be given by:
n.d(p l. ,T)
c n*(p ) + 11(0,T)
c (78)
The combination of Eqs. (76)-(78) results in Eq. (75).
For xenon, the reference temperature is chosen as 30°C, the highest temperature for which viscosities were measured. The low den- sity viscosities are taken from the data of Clarke and Smith[52
J and are reproduced in Figure 34 for the range of interest. In Eq. (75), n(O,T f) = 234.4 micropoise and the values for n(O,T) range from
re
226.8 to 224.4 micropoise for the states considered.
The critical point of the xenon used in the present investigation has been determined by visual observation (see Appendix II). The values are Tc= 16.627 ±0.005°C and pc = 1.11± 0.01 g/cm3
. These compare with the values of T = 16.590°C and p = 1.105 g/cm3
as reported by
c c
Weinberger and Schneider[53
J.
Their critical temperature is presumably reported in relation to the 1948 ITS. This would be equivalent to 16.584°C on the 1968 IPTS.The isochore closest to the determined critical density is #5 at p
=
1.1092 g/ cm3. This is taken as the "critical" isochore and used in Eq. (75). The value of n(p T ) is 535.2 micropoise.c' ref
Using the data given in Table IX, the anomalous viscosities have been calculated and are listed in Table XI. The onset of the critical behavior is around T-T
c
0 -2
= 3.5 C, £ = l.2 x1Q The closest approach to critical is at T -T
=
0.034°C, £ = 1. 2 x 10-4. At temperatures c
closer to critical than 0.034°C, it was not possible to get sensible viscosity measurements due to the severe density gradients set up in the gravitational field. Gravitational effects are discussed in a later section. The maximum value for the anomalous viscosity at the closest approach to critical is about 15% of the ideal viscosity.
In Figure 35, the logarithm of the anomalous viscosity is plotted as a function of the logarithm of the reduced temperature difference from critical. As can be seen, the data do not fall on a straight line and do not seem to be consistent with the idea of an exponential divergence. In Figure 36, the anomalous viscosity is plotted as a function of the logarithm of reduced temperature differ- ence. The data can be fitted to a straight line of equation
T -T
-37.71 log(-T-__£) - 68.66; t.n in micropoise c
(79)
by a linear least-squares technique. The error bars in Figures 35 and 36 represent the estimated precision of a viscosity measurement, ±0.5%.
The results seem to be consistent with the idea of a logarith- mic viscosity divergence, the form of which is given in Eq. (4). The straight line fit is only weakly dependent upon the nature of the anomalous viscosity determination and a straight line
ably well fitted to a plot of actual viscosity versus
could be reason- T -Tc
log( T ) c The difficulty of distinguishing among a logarithmic divergence, a small exponential divergence, and a finite limit for viscosity has been previously mentioned. The results of the present investigation should not be construed so as to preclude the possibility of a finite limit or small exponential divergence for the anomalous viscosity.
The critical point of ethane has also been determined by visual observation (see Appendix II). The values were found to be T = 32.218 ± 0.005°C and
c
p = 0.2055 ± 0.002 g/cm 3 .
c There is a dis·-
parity in the critical parameters for ethane as reported in the literature. A recent investigation by Miniovich and Sorinar54J gave
the values of T
c 32.200 ± 0.005°C and Pc 0.2041± 0.0007 g/cm3. The temperature scale was not given.
The isochore closest to the determined critical density is #2 at P = 0.20952 g/cm 3 . This is taken as the "critical" isochore and used in the determination of the anomalous viscosity.
The reference temperature for ethane is chosen as 49 C, t0 he highest temperature for which viscosities were measured. The value of n(p ,T f) is 200.2 micropoise. The low density viscosities are taken
c re
from the data of Carmichael and Sage[50J and are reproduced in Figure 37 for the range of interest. The range in n(O,T) is from 95.7 to 95.5 micropoise and n(O,T re f)
=
100.4 micropoise.Using the data given in Table X, the anomalous viscosities have been calculated and are listed in Table XII. The onset of the
critical behavior is around T - T c closest approach to critical is at
0 -3
0. 8
c '
£=
2 . 5 x 10 . T - T = 0.017°C ,c
Again gravitational problems are blamed for the inability to more The
closely approach the critical temperature. The maximum value for the anomalous viscosity at the closest approach to critical is about 16%
of the ideal viscosity.
Figures 38 and 39 show the logarithm of the anomalous viscosity and the anomalous viscosity plotted as a function of the logarithm of reduced temperature difference. The error bars represent the esti- mated precision of the measurements, ±0.5%. The data in Figure 39 are fitted to a straight line of equation
T -Tc
-15.15 log( T ) - 32.57; 6n in micropoise c
(80)
by a linear least squares technique. The ethane data also seem to be consistent with the idea of a logarithmic divergence of viscosity as given in Eq. (4).
B. Gravitational Effects
A fluid of finite height in a gravitational field is com- pressed under its own weight. Close to the critical point where the isothermal compressibility becomes very large, this phenomenon results in large density gradients in the direction of the field. These
effects are well known[SS] and are of great hindrance to critical region studies, the present investigation not excepted.
It is possible to estimate the density gradients in the cell by using the scaling law equation of state. The scaling law equation of state is a universal equation which can be used to predict the reduced PVT properties of fluids in the critical region as follows[Sl] :
where
6p*
x
µ
Sand 6
6µ*
=
6p*j6p*l6-l h(x) µ(p,T) - µ(pc,T)p - p c
E
p /p c c
is the chemical potential
are the critical exponents, and h(x) is a function only of x .
(81)
The following form of h(x) has been proposed by Vicentini-Missoni and Levelt Sengers[56l:
h(x)
x+x x+x 26[6(6-1) - 1) 2B
El ( x o)[l+E2( x o) ] (82)
0 0
Vicentini-Missoni and Levelt Sengers give the following values for the parameters, derived from the available xenon PVT data in the
. . 1 . [56) cr1t1ca region :
B 0.350
0 4.6
x 0.186
0
El 2.96 E2
=
0.37p c 57.636 atm
From Eqs. (81) and (82) the difference in chemical potential µ(p,T) - µ(pc,T) can be calculated for a given density and tempera- ture. The chemical potential difference is proportional to height ln the gravitational field, as follows[5lJ:
z - z c
where g is the local acceleration of gravity
and z is the height at which the local density equals the c
critical density.
(83)
Using the values of T
c and from the present investigation, the density profile has been calculated for T - T
=
0.034°c, using Eqs.c
(81)-(83). This temperature difference represents the closest
approach to the critical temperature of xenon in the present investi- gation. The calculated profile is shown in Figure 40. It should be
made clear that the curve in Figure 40 does not necessarily represent the density profile in the cell, but simply shows the variation of density with height, with the position of the height coordinate not specified. In order to specify the position of the vertical compon- ent, it is necessary to know the value of the local density at some height in the cell.
In the present investigation, the viscosity measurements can potentially be affected by the density gradients in two ways. First, close to the critical temperature the density can vary appreciably over the vertical component of the crystal, even though the crystal is aligned parallel to the horizontal and the diameter is only 0.3 cm. If the density at the center of the crystal is the critical density, it can be seen from Figure 40 that for T - T
=
0.034°cc the density
can vary by ±1. 5% over the ±0.15 cm of the crystal diameter. If the density at the center of the crystal is different from critical, the variation is not as large, but is still quite substantial at densi-
ties close to critical. As the critical temperature is more closely approached, the gradient over the crystal increases rapidly. It is difficult to predict what effect the density gradient would have on the measured viscosity. Presumably some sort of "averaging" of the viscosity-density product would occur, possibly masking the anomalous
increase. The crystal, of course, measures the viscosity-density product only of a very thin layer of fluid surrounding the crystal, and is unaffected by the bulk fluid.
The second problem is closely related to the first. At an average density close to, but not exactly at the critical, the densjty
at the level of the crystal changes as the critical temperature is approached, and can become quite different from the average density.
Since the average density value is used in calculating the viscosity from the resonant properties of the crystal, what results is an
apparent viscosity for a non-existing state. This, of course, can be quite different from the real viscosity for the existing state. The cell is about 1.8 cm high, with the crystal positioned approximately in the center.
It is believed that the above-mentioned difficulties caused the apparent viscosity values to degenerate for values of
T - T < 0.034°C for xenon. The ethane problems were, of course,
c
similar to those found for xenon. Since ethane is quite a bit less dense than xenon, the gravitational effects might not be expected to be as severe. Indeed, the viscosity values seemed to be sensible up to T - T = 0.017°C.
c